Piezo-actuated MEMS resonator with surface electrodes

ABSTRACT

A microelectromechanical system (MEMS) resonator includes a degenerately-doped single-crystal silicon layer and a piezoelectric material layer disposed on the degenerately-doped single-crystal silicon layer. An electrically-conductive material layer is disposed on the piezoelectric material layer opposite the degenerately-doped single-crystal silicon layer, and patterned to form first and second electrodes.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a divisional of U.S. patent application Ser. No.15/627,029 filed Jun. 19, 2017, which is a divisional of U.S. patentapplication Ser. No. 14/617,753 filed Feb. 9, 2015 (now U.S. Pat. No.9,705,470), which claims priority to U.S. Provisional Patent ApplicationNo. 61/937,601 filed Feb. 9, 2014. Each of the above-identified patentapplications is hereby incorporated by reference herein.

TECHNICAL FIELD

The disclosure herein relates to the field of microelectromechanicalsystems (MEMS).

INTRODUCTION

Resonators are a fundamental component of modern technology, commonlyused in timing, sensing, and signal processing applications. Devicessuch as oscillators, sensors, actuators, filters, etc. may beconstructed in whole or in part using resonant structures.

A resonator may be viewed as a passive system for which the linearresponse function exhibits several oscillatory cycles before decayingsubstantially. Two principal parameters that describe a resonator arethe oscillation frequency of the linear response function, known as thenatural frequency, and the rate of decay of the linear responsefunction. The natural frequency divided by twice the exponential decayrate is known as the quality factor, which is an important metric ofresonator caliber.

Frequency stability over temperature is an important characteristic formany applications. For example, the 802.11b wireless communicationstandard requires frequency stability of better than +/−25 ppm over aparticular temperature range, such as from −40 C to +85 C for industrialapplications.

The variation in natural frequency of a resonator as a function oftemperature can be described by a series expansion about a specifiedreference temperature T₀,

$\begin{matrix}{{{f(T)} = {f_{0}{\sum\limits_{n = 0}^{\infty}\;{\lambda_{n}\left( {T - T_{0}} \right)}^{n}}}},} & (1)\end{matrix}$where T is the ambient temperature, f₀ is the nominal resonantfrequency, λ_(n) is the n^(th) or n^(th)-order temperature coefficientof frequency (TCF), and λ₀ is 1 by definition. The first TCF is alsocalled the linear TCF, the second TCF is also called the quadratic TCF,the third TCF is also called the cubic TCF, etc. The deviation infrequency Δ_(f) between the frequency at the ambient temperature and thenominal frequency is commonly expressed in units of parts per million(ppm). The value of Δ_(f) is determined by the magnitudes and signs ofthe various TCF terms and the difference between the ambient temperatureand the reference temperature.

Although techniques have been developed to null the first-order TCF ofmicromechanical resonators using a composite mechanical structure,first-order TCF nulling is increasingly insufficient to meet targetfrequency stability ranges, particularly in emerging applications. Also,while silicon microstructures with silicon dioxide (SiO₂) and germaniumdioxide (GeO₂) coatings have been proposed for multi-order temperaturecompensation, such structures tend to bring additional operational andfabrication challenges. SiO₂ layers can increase bulk acoustic losses(thereby degrading the quality factor), interfere with the desired modeshape, reduce transduction effectiveness, produce undesired hystereticeffects (e.g., charging) and so forth. Moreover, the quadratic residualof frequency with some compensation materials can be unsuitably large.More generally, the mere proliferation of material layers (i.e., astypically required by proposed multi-order temperature compensationschemes) may degrade resonator performance, as energy losses at thematerial interfaces during resonance tends to lower the resonatorquality factor and interfere with the desired mode shape. Fabricationcomplexity and cost also tend to rise quickly with increased layercount.

BRIEF DESCRIPTION OF THE DRAWING

The various embodiments disclosed herein are illustrated by way ofexample, and not by way of limitation, in the figures of theaccompanying drawings and in which like reference numerals refer tosimilar elements and in which:

FIG. 1A illustrates examples of positive/negative first-order andsecond-order temperature coefficients of frequency (TCFs), and alsoshows positive and negative 0^(th) order TCFs (i.e.,temperature-independent frequency offsets);

FIGS. 1B-1E illustrate embodiments of a resonant structure consisting ofone or more materials that allow for control of first- and higher-orderresonator temperature coefficients of frequency;

FIG. 1B presents a top-down view of a resonator having a degeneratelydoped semiconductor (DDS) layer, with dashed lines indicating possibleelectrode configurations on and off the resonator (such electrodes maybe patterned) and shown partitioned on the resonator into twoside-by-side electrodes (many alternative patterns are possible);

FIGS. 1C-1E show cross sections of the structure through the dashed linesegment A-A′;

FIG. 1C illustrates a resonator embodiment constructed of a singlematerial capable of controlled TCF engineering such as a degeneratelydoped semiconductor;

FIG. 1D shows the addition of a second layer that might serveelectrical, mechanical, or other functions;

FIG. 1E shows a three layer resonator embodiment where the second layermight be a piezoelectric material and the third layer might be aconductive material, possibly patterned and in which electricalconnections can be made to the structure in a variety of ways (anon-exhaustive list of example electrical connections are labeled aselectrodes or electrode regions);

FIG. 1F illustrates an embodiment of a resonator having a degeneratelydoped semiconductor layer in which dopant concentration and/or type isnon-uniform across the resonator body;

FIG. 1G illustrates an embodiment of a resonator having regions in or onthe resonator of a locally deposited or patterned material with desiredproperties;

FIG. 1H illustrates examples of engineered TCFs within resonatorembodiments of FIGS. 1B-1G and other embodiments disclosed herein;

FIG. 1I illustrates an embodiment of a three-layer resonator thatincludes a degenerately doped single-crystal silicon layer coated inaluminum nitride (to form a piezoelectric layer) and an additionalconductive layer (e.g., a metal layer or another degenerately dopedpolysilicon electrode layer);

FIG. 1J illustrates the resonator embodiment of FIG. 1I in which theconductive layer is made sufficiently thin to render its contribution tothe TCFs of the composite structure negligible;

FIG. 1K illustrates an embodiment of a MEMS resonator havingdegenerately doped single-crystal silicon layered with aluminum nitrideand with those two layers sandwiched between top and bottom electrodes;

FIG. 2 illustrates an embodiment of a MEMS system in which a resonatoris co-fabricated on a single substrate with a temperature sensitiveelement and a heater (the temperature sensitive element and/or theheater can be optionally excluded in alternative embodiments);

FIGS. 3A-3E illustrate embodiments of MEMS systems in which atemperature-stable MEMS resonator is combined with additional activetemperature compensation for improved frequency stability overtemperature;

FIG. 3A illustrates an embodiment of a MEMS resonator system in whichthe oscillation frequency of a resonator and sustaining circuit(combined to form an oscillator) is modified as a function oftemperature;

FIG. 3B illustrates a more detailed embodiment of the system in FIG. 3Ain which resonator frequency is tuned using an electrostatic fieldgenerated by control electrodes;

FIG. 3C illustrates a more detailed embodiment of the system in FIG. 3Ain which resonator frequency is tuned through static mechanical stressusing an electrostatic field within a piezoelectric material;

FIG. 3D illustrates a more detailed embodiment of the system in FIG. 3Ain which a tunable capacitive load is provided within the oscillator(and more specifically within a sustaining circuit) in series with theresonator.

FIG. 3E illustrates an alternative approach to temperature compensationthat effects a temperature-dependent frequency modification of theoscillator output;

FIG. 3F illustrates an MEMS resonator system in accordance with FIG. 3Eand in which a temperature-dependent signal is be provided to afractional-N phase locked loop to generate an output frequency that issubstantially more stable over temperature than the raw output of theresonator and sustaining circuit;

FIG. 3G illustrates programmable/storage circuitry that may be employedin connection with resonator system embodiments of FIGS. 3A-3F to storeTCF coefficients, control settings or other information useful forcontrolling resonator operation.

FIG. 3H illustrates an embodiment of a MEMS resonator die having a DDSresonator and a resistive element that is laser-trimmed (or otherwiseset or established) to indicate one or more values relating to behaviorof the MEMS resonator over temperature;

FIG. 3I illustrates a MEMS resonator system that employs a programmablestorage as shown in FIGS. 3G and/or 3H;

FIG. 4A illustrates an embodiment of a degenerately-doped-silicon MEMSresonator integrated with one or more components of an activetemperature compensation system;

FIG. 4B illustrates an example of a temperature compensation operationwithin the embodiment of FIG. 4A;

FIG. 5A illustrates an embodiment of a DDS resonator having one or morefeatures described in connection with FIGS. 1B-4B;

FIG. 5B illustrates an exemplary cross-section of the DDS resonatorshown in FIG. 5A; and

FIG. 5C illustrates an exemplary finite-element model of the DDSresonator of FIG. 5A.

DETAILED DESCRIPTION

In various embodiments disclosed herein, the material properties of adegenerately doped semiconductor (DDS) and the manner of its deploymentwithin a monolithic or composite resonant structure are engineered tocontrol the linear TCF and at least one higher order TCF of a resonantmode of the structure, thus enabling increased control over thefrequency-temperature relationship of the resonant structure. Forexample and without limitation, the semiconductor material chosen, itscrystal size and orientation, dopant type(s) and concentration(s),resonator geometry (including resonance mode shape and orientationrelative to the crystal axis), may all be parameterized within a “DDSresonator” design and thus used as adjustable “knobs” for TCFengineering.

In other embodiments, a composite structure, of which at least oneconstituent material is a degenerately doped semiconductor, isengineered to control the linear TCF and at least one higher order TCFof a resonant mode of the structure. In addition to methods ofengineering monolithic resonators, in composite resonators, the geometryand orientation of various component materials and their relation to theresonance mode shape may be adjusted for TCF engineering. One example ofsuch a composite structure includes a degenerately doped semiconductormaterial and a piezoelectric material and is engineered to control thelinear TCF and at least one higher order TCF of a resonant mode of thestructure. As a more specific example, a composite structure includingdegenerately doped silicon (an example of a DDS) and aluminum nitride (apiezoelectric material) is engineered to achieve a resonance mode forwhich the first- and second-order TCFs are both substantially zero orwithin a specified tolerance of zero over a predefined or programmedtemperature range.

In yet other embodiments, one or more resonator layers or regions madefrom a degenerately doped semiconductor serves at least two functions:enabling one or more TCFs of a resonant mode to be specificallyengineered, and serving as an electrical conductor within the resonator.In a number of implementations, two or more electrically-isolated DDSlayers of a resonant structure are applied in such a multi-functioncapacity, contributing to engineering of one or more TCFs of thecomposite structure and serving as respective electrical conductors.

In further embodiments, a DDS resonator as described above may beintegrated with one or more components of an active temperaturecompensation (ATC) system that maintains the DDS resonator at/within aprogrammed or predefined temperature or range of temperatures (e.g., fordiagnostic, measurement, calibration, operational control or otherpurposes). A combined DDS resonator and ATC system (whether integratedor not) may be designed to generate an output signal with improvedtemperature stability compared with a passively-compensated resonatoralone.

The improved temperature stability of a DDS resonator may enable moreresolute or fine-grained active frequency-compensation approaches thatmay not be feasible with a less temperature-stable MEMS resonator (e.g.,electrostatic tuning that achieves relatively small tuning range).Examples of frequency-compensation approaches that may become viable (ormore effective, efficient, etc.) through use of more temperature-stableDDS resonators include electrostatic control electrodes, capacitiveloading in series with the resonator, mechanical stress tuning using apiezoelectric material, and fractional-N phase locked loops, to name afew.

Multi-Order TCF Compensation

In contrast to many conventional temperature compensation schemes,TCF-compensating embodiments disclosed herein compensate not only forthe first-order TCF, but also higher-order TCF terms (e.g., thequadratic and cubic TCFs)—a more complex “multi-order” compensationwhich turns out to be important for many applications. Examples of suchapplications, include (without limitation) resonators engineered to:

-   -   null or otherwise attenuate (e.g., to zero, substantially zero,        or an otherwise negligible level) the first, second, third, and        fourth order TCF terms or any two of those.    -   exhibit a nonzero value of the linear TCF specifically chosen to        partly or wholly compensate (or cancel or counteract) the cubic        TCF and thereby reduce the absolute frequency variation over a        particular temperature range.    -   exhibit one or more local extrema in the temperature-dependent        frequency (i.e., having a temperature at which a local minimum        or maximum frequency occurs), also called “turn-over        temperatures.”    -   exhibit a turn-over at the nominal operating temperature.

Also, in a departure from TCF compensation schemes that cumulativelyapply material layers to compensate respective TCF terms (e.g., one ormore material layers to compensate for first-order TCF, one or moreother material layers to compensate for second-order TCF, etc.),embodiments of temperature-engineered resonant structures disclosedherein include a semiconductor layer or region engineered to compensate,by itself, for both the linear TCF and one or more higher-order TCFs andthus enable construction of “short stack” MEMS resonators—resonatorsconsisting of a reduced number of material layers relative tolayer-per-TCF implementations. For one example, the linear TCF of aresonator fabricated from an anisotropic degenerately dopedsemiconductor, such as single-crystal silicon, is adjusted by rotatingthe orientation of the resonator geometry relative to underlying crystalaxes. As a second example utilizing an anisotropic degenerately dopedsemiconductor, the linear TCF of a resonator is tuned by altering thegeometry of the resonator and/or the mode shape. As another example, ina number of embodiments, a semiconductor material is doped to asufficiently high concentration that the polarity of the second-orderTCF (i.e. the sign of the quadratic TCF) of a resonator constructed fromthat material is reversed relative to the second-order TCF of aresonator constructed from a more lightly doped version of thatsemiconductor (e.g., yielding a positive quadratic TCF, in contrast tothe negative quadratic TCF exhibited at lower dopant concentrations).The reverse-polarity quadratic TCF may be exploited to partially orwholly cancel the quadratic TCF of one or more other materials (forexample a piezoelectric material having a quadratic TCF polarityopposite that of a degenerately doped semiconductor material) within acomposite resonator embodiment over a given temperature range, thusproducing a resonator with a reduced quadratic TCF magnitude. Asexplained, a resonant structure with engineered linear and higher orderTCFs can be fabricated exclusively from a degenerately dopedsemiconductor, or degenerately doped semiconductors may be deployedwithin composite resonant structures to enable engineered temperaturestability. For example, composite structures suitable for piezoelectricmicromechanical resonators are disclosed below and address the problemsof frequency stability over temperature and the engineering of linearand higher-order TCFs. As discussed, such composite structures offerhigh quality factor, low hysteresis over temperature, low aging offrequency over time, compatibility with batch micro-fabrication methods,small form factor, insensitivity to shock and vibration, and otherdesirable features.

Resonant structures as described herein may be fabricated wholly orpartly from a degenerately doped monocrystalline or polycrystallinesemiconductor (e.g., silicon, germanium, diamond, carbon, siliconcarbide, compound semiconductors such as silicon germanium, etc.) orusing combinations of such semiconductors. Pure and lightly-dopedsemiconductors are insulating materials at low temperature. When thesemiconductor is doped with certain impurity atoms, above some dopantconcentration the semiconductor will exhibit metallic or highlyconductive behavior even at low temperatures (e.g. for single crystalsilicon, approaching 0 Kelvin). Such a semiconductor is said to be“degenerately doped”. For example, in single-crystal silicon, this mayoccur at a phosphorus doping level at or above 4E18 atoms/cm³. Moregenerally, the onset of degeneracy varies with semiconductor and dopant.In addition to a change in electrical conductivity, various materialproperties of semiconductors change with doping level, as well as thetemperature dependencies of various material properties. In particular,resonators fabricated with a degenerately doped semiconductor canexhibit significantly different TCFs than similar resonators constructedwith a more lightly doped version of the semiconductor material. In somecases, it is even possible to reverse the sign (or polarity) of one ormore TCFs by altering the doping level of a semiconductor used as astructural material in a resonator. These temperature-dependency changesare exploited in a number of resonator embodiments to enablesophisticated, targeted engineering of temperature coefficients.

In specific embodiments of resonant composite structures disclosedherein, two of the constituent materials are a degenerately dopedsemiconductor (DDS) and a piezoelectric material. This compositestructure, an example of a DDS resonator, can be engineered tosimultaneously achieve target values or ranges of values for two or moreof the TCFs of a particular resonance mode. In one embodiment, forinstance, a composite structure is constructed from degenerately dopedsilicon (a semiconductor) and aluminum nitride (a piezoelectricmaterial) and engineered such that the first-order and second-order TCFsof a particular resonance mode of the structure as a whole (i.e., thelinear and quadratic TCFs, respectively) are both within a specifiedtolerance from zero, thus yielding a temperature-insensitive resonatoror temperature-compensated resonator.

Examples of the piezoelectric materials include but are not limited toaluminum nitride, zinc oxide, quartz or lead zirconate titanate. Inaddition to the two primary constituent materials, additional materialsmay be present in the composite structure. In particular, anotherconductive material such as metal or another semiconductor layer (whichmay also be degenerately doped) may also be included to serve as anadditional electrode (as discussed below, the DDS layer may serve asanother electrode within the composite structure). Examples of suitableelectrode materials include but are not limited to heavily dopedsilicon, silicides, tungsten, molybdenum, titanium or aluminum. The termelectrode is used herein to mean, without limitation, a conductivematerial used to establish an electric field for transduction ofelectrical to mechanical energy or mechanical to electrical energy. Notethat layers applied as electrodes may also serve other functions, forexample and without limitation, a piezoresistive function, a heatingfunction, etc.

The embodiments disclosed herein address many or all of theaforementioned problems and issues for resonator performance throughengineering multiple parameters or design degrees of freedom of astructure with reduced sensitivity to temperature while providingpiezoelectric coupling for some structures, high quality factor, lowhysteresis, low aging, batch micro-fabrication compatibility, small formfactor, insensitivity to shock and vibration, etc.

Mechanical Resonators

For mechanical resonators, the natural frequency is determinedprincipally by the mass density and the stiffness of the material fromwhich the resonator is constructed. The change in material stiffness dueto change in ambient temperature is of principal concern in the designand manufacture of resonators because it changes the natural frequencyof the resonator. The change in material geometry due to thermalexpansion or contraction is also a concern as it also changes thenatural frequency of the resonator.

For pedagogical purposes, it can be illustrative of the principle of TCFengineering to examine a simple model. One such model is where theresonator structure is constructed of thin layers and only the motion ofthe material on a single axis is considered. For this simple example,the stiffness of a material is described by a single number, theeffective elastic modulus. This is a simplification of the physical casewhere all dimensions of motion should be considered and the stiffness ofa material might be described by a tensor. In the simple model, theelastic modulus of an anisotropic material depends on the materialorientation.

For composite structures, a simple model for the n^(th) temperaturecoefficient of frequency is a weighted average of the contributions ofall constituent parts of the resonator. This average can be written asfollows for a stack of thin films (or layers) of material:

$\begin{matrix}{\lambda_{n} = \frac{\sum\limits_{i}\;{\lambda_{n}^{(i)}E_{i}t_{i}}}{\sum\limits_{i}\;{E_{i}t_{i}}}} & (2)\end{matrix}$where the sum over i is taken over all films in the stack, t_(i) is thethickness of the i^(th) film, E_(i) is the elastic modulus of the i^(th)film, and λ_(n) ^((i)) is the n^(th) material TCF for the i^(th) film.The coefficient λ_(n) ^((i)) is a material parameter of the i^(th) filmthat lumps together the contributions from thermal expansion andtemperature-sensitivity of the elastic modulus to give the temperaturecoefficient of frequency for a resonator constructed of that materialalone. Equation (2) can be generalized for composite structures witharbitrary geometries, non-uniform, and anisotropic materials. Note that,in that case, the λ_(n) ^((i)) coefficients can be tensors. The n^(th)TCF for a resonator constructed out of at least one material is λ_(n).

Equation (2) shows that the first requirement for engineering thetemperature coefficients of a resonator's frequency using compositematerials is to use materials that bracket the desired values. Forexample, if λ_(n) is desired to be zero, this result can be obtained ifat least one λ_(n) ^((i)) is positive and at least one other λ_(n)^((j)) is negative.

Equation (2) also indicates that in order to simultaneously engineer Ntemperature coefficients such that λ_(n)=λ_(n)*, where λ_(n)* is thedesired value of the n^(th) coefficient, equation (2) can be split intoN separate equations. Typically the solution of N equations is obtainedwith the variation of at least N variables. These N variables can becalled design variables and they should have sufficient design authoritysuch that the solution to these N equations is in an accessible space.Design authority is a description for the magnitude of the effect that achange in a given design parameter has on a metric of interest. Theaccessible design space depends on fabrication constraints (e.g.material selection, film thickness ranges and control accuracy) andoperation constraints (e.g. quality factor, frequency, motionalresistance).

Despite the traditional emphasis on linear TCF control, analysis showsthat composite resonator performance may be substantially increasedthrough control over at least the first two temperature coefficients. Asmentioned above, a resonator implementation that exhibits control overat least the first two temperature coefficients of frequency can beconstructed from aluminum nitride and degenerately doped single crystalsilicon. Such a resonator can be compatible with piezoelectrictransduction and may exhibit other advantageous properties including butnot limited to high quality factor, low hysteresis over temperature, lowfrequency aging over time, batch micro-fabrication compatibility, smallform factor, and insensitivity to shock and vibration.

In addition to controlling at least the first-order and second-orderTCFs, the following non-exhaustive list of criteria were applied indesigning selected resonator embodiments disclosed herein:

-   -   The temperature coefficients of the individual materials combine        to yield the target temperature coefficient of frequency for the        overall structure. For example, if the target first-order TCF is        at or near zero, and the first-order TCFs of at least one        component-material is substantially positive, then the        first-order TCF of at least one other component-material is        engineered and/or selected to be negative.    -   There are N design parameters to enable control over N        temperature coefficients.    -   The combination of design parameters have sufficient design        authority to enable a solution within the design space defined        by fabrication constraints and design constraints.

Aluminum nitride resonators typically have negative linear and quadraticTCFs. The temperature coefficients of thin film polycrystalline aluminumnitride depend weakly on the film structure. Similarly, resonatorsconstructed from non-degenerate single-crystal silicon have negativelinear and quadratic TCFs, and the TCFs tend to be only weakly dependenton crystal orientation.

The linear TCF of a resonator constructed from degenerately dopedsingle-crystal silicon can be positive or negative depending on crystalorientation, doping level and mode shape. Thus, crystal orientationconstitutes a design parameter (or design degree of freedom) that may beadjusted to control the linear TCF term. The quadratic TCF of aresonator constructed from degenerately doped single-crystal silicon canbe positive or negative depending on dopant concentration, crystalorientation, doping level and mode shape. FIG. 1A illustrates examplesof such positive/negative first-order and second-order TCFs, and alsoshows positive and negative 0^(th) order TCFs (i.e.,temperature-independent frequency offsets).

It becomes possible to independently tune (i.e., control and potentiallynull) both the first-order and second-order (linear and quadratic) TCFcharacteristics of a degenerately doped single crystal silicon layerthrough manipulation of orthogonal design parameters, for example,crystal orientation and dopant concentration. Polycrystalline siliconresonators can be degenerately doped to achieve a range of linear andquadratic TCF values, although if the material lacks a dominantcrystallographic orientation then one design degree of freedom may belost.

FIGS. 1B-1E illustrate exemplary embodiments of a DDS resonator with aplan view of the resonator shown at 1 a together with optional electrodedispositions within and adjacent the resonant structure.

Referring first to FIG. 1B, a resonant structure including at least onedegenerately doped semiconductor layer (“DDS Resonator”) is disposedbetween two electrode structures (e.g., used for driving and sensing theresonator, respectively) and having one or more anchor points. In theembodiment shown, beams extend from opposite sides of the resonator bodyto establish dual anchor points, though more or fewer anchor points mayapply in alternative implementations. Also, while an oblong orrectangular resonator shape is depicted, DDS resonators may befabricated in any practicable shape according, for example, toapplication demands.

The electrodes on either side of the DDS resonator are shown in dashedoutline to emphasize their optional nature (a convention applied inother drawings herein, though the absence of dashed-line presentationshould not be construed as meaning that a given element or structure isrequired). Also, one or more electrically-isolated electrode regions maybe implemented within a given resonator layer as shown by the T-shapedregions outlined within the resonator body. One or morealternately-shaped electrode regions (i.e., having different shapes thanthose shown) may be employed in alternative embodiments, and theresonator body itself may also be employed as an electrode.

In the case of a single-layer degenerately doped silicon resonator, themotion of the resonator can be sensed electrostatically orpiezoresistively with appropriate electrical connections. FIGS. 1C-1Eillustrate cross-sections of exemplary material stacks (i.e., across theresonant structure at line A-A′ in FIG. 1B) including a monolithic(single-layer) stack at 1C, two-layer material stack at 1D andthree-layer material stack at 1E. As explained below, additionalmaterial stacks and/or stacks of different material than shown in FIGS.1C-1E may be present in alternative embodiments. As discussed above,because a single layer of uniformly degenerately doped silicon possessesat least two design parameters (crystallographic orientation and dopantconcentration), a single-layer resonator (FIG. 1C) composed of uniformlydegenerately doped single-crystal silicon can be engineered to havefirst-order and second-order TCFs that are equal or near to zero throughthe combination of fabrication process and design. Moreover, the dopantconcentration need not be uniform. This allows for an arbitrarily largenumber of design parameters (i.e., “knobs” or degrees of freedom formanipulating resonator performance through design). It may also beadvantageous to create one or more regions in or on the resonator thathave different dopant concentration and/or different dopants as shown,for example, by the different degrees of shading in FIG. 1F. As shown inFIG. 1G, regions in or on the resonator could also be created with alocally deposited or patterned layer of material with desiredproperties. Locating these regions in areas of high stress in thevibrational mode shape of the resonator, for example, may beadvantageous, enabling control over first, second, third, or higherorder TCF values.

The resonator can also be engineered to have non-zero but controllablefirst- and second-order TCFs in order to accomplish specific designintent. For example, the first-order TCF can be designed such that itminimizes the total frequency variation over temperature by compensatingfor third-order and other odd-numbered higher order TCFs. As anotherexample, the first-order TCF can be designed in order to adjust thetemperature at which the frequency change with respect to temperaturereaches an inflection point, a local minimum or a local maximum. Thetemperature at which the resonator reaches a local minimum or maximumfrequency is commonly referred to as a turnover temperature. Theresonator TCF may also be designed to cancel the TCF associated with itssustaining circuit (i.e. a circuit that sustains the mechanical motionof the resonator) or oscillator system. As a final example, the first-and second-order coefficients can be selected such that they arerelatively insensitive to angle and dopant concentration for improvedmanufacturability. FIG. 1H illustrates examples of such engineered TCFs.

Referring again to FIG. 1D, a two-layer resonator composed ofdegenerately doped silicon and an additional film can be constructed. Ifthe silicon layer is single crystal then this structure possesses atleast three design parameters: crystallographic orientation and dopantconcentration as described above, and additionally the ratio of thesilicon thickness to the thickness of the added film. Thus, a two-layerresonator composed of degenerately doped silicon and an additional filmmay be capable of controlling three TCFs. In a number of embodiments,including that shown in FIG. 1c , the additional film is a piezoelectricmaterial (e.g., aluminum nitride), though the additional film (or othermaterial layer) may alternatively be any semiconductor, insulator ormetal material selected for its TCF coefficients or another desirablemechanical or electrical property.

Additionally, a three-layer resonator can be formed as shown in FIG. 1E(i.e., with or without electrically isolated electrode regions asdiscussed). In one embodiment, shown for example in FIG. 1I, such athree-layer resonator includes a degenerately doped single-crystalsilicon layer that is coated in aluminum nitride (to form apiezoelectric layer) and an additional conductive layer (e.g., a metallayer or another degenerately doped polysilicon electrode layer). Asshown by the TCF control diagram in FIG. 1I, the first- and second-orderTCFs can be controlled by varying the crystal orientation of the bottom(single-crystal) silicon at a particular doping level and thickness, thealuminum nitride thickness, and the polysilicon thickness at aparticular dopant concentration. The preferred crystal orientation inthe polysilicon film may also affect the first- and second-order TCFs.An alternative set of design parameters can be chosen in order toengineer the TCF of the film stack, and the previous example is just oneof many possibilities. Additionally, as shown at FIG. 1J, one of thelayers in a three-layer resonator can be made sufficiently thin that itscontribution to the TCFs of the composite structure is minimal (e.g.,negligible or otherwise attenuated) and the compensation problem reducesto that of the two-layer resonator case. For example, the top layer maybe implemented by a thin conductive metal or semiconductor layer insteadof a degenerately doped polysilicon electrode layer.

Additional layers can be added to the stack. One example, shown forexample in FIG. 1K, includes a degenerately doped single-crystal siliconwith aluminum nitride sandwiched between a top electrode and bottomelectrode. The electrodes can be manufactured from any conductive metalor semiconductor film, for example polysilicon, aluminum, molybdenum,titanium, tungsten, or a silicide formed using a metal and silicon.

In a number of embodiments, including those shown in FIG. 1B-1E,multiple degrees of freedom are applied to compensate multipletemperature coefficient orders. When degenerately doped silicon is usedfor temperature compensation it can provide control over the first- andsecond-order terms. Degenerately doped single crystal siliconincorporates at least two degrees of freedom into a single layer:crystal orientation and dopant concentration. In principle, a thirddegree of freedom is available once a piezoelectric film is deposited onthe highly doped single crystal silicon determined by the thicknessratio of the films. This additional design authority is illustrated inFIG. 1I.

Degenerately doped silicon can replace the function of multiple filmsthat are otherwise used in temperature compensated piezoelectricresonators including, for example and without limitation, oxidetemperature compensation layers and metal electrode layers. The sheetresistance of silicon is a function of its thickness and carrierconcentration, and degenerately doped temperature compensation layershave sufficiently low sheet resistance (e.g., less than 1 ohm/square) toprovide low electrical impedance. Material interfaces in compositeresonators tend to introduce mechanical dissipation and the potentialfor a degradation of frequency hysteresis over temperature and frequencyaging, and thus, though counterintuitive in view of functional benefitsthat may result from additional resonator layers, a reduction in thenumber of resonator layers tends to enhance engineering control offrequency stability over temperature.

In various embodiments, a single MEMS system may include multipleelements co-fabricated on the same substrate as a DDS resonator.Referring to FIG. 2, for example, a MEMS system 200 may include a DDSresonant element 201 (i.e., one form of MEMS resonator), one or moretemperature sensing elements 203 (“temperature sensor”), and one or moreheating elements 205 (“heater”). The temperature sensitive element maybe used as part of an active temperature compensation system. Oneexample of a temperature sensing element is a thermistor, which has atemperature-dependent electrical resistance. A heater may be includedfor initial calibration of the resonator frequency stability or tomaintain the MEMS system at an approximately constant temperaturedespite variation in the ambient temperature. The temperature sensorand/or heater may be optionally excluded from the MEMS system. Asdescribed above, the resonator structure may be used as a sensor insteadof a frequency reference. Examples of alternative resonator applicationsinclude filters, gyroscopes, accelerometers, pressure sensors,microphones, magnetometers and mass sensors.

DDS resonators as described herein may be deployed with or withoutsupplemental temperature compensation, thus effecting a purely passivetemperature compensation scheme, or a combination of, for example,passive and active temperature compensation. Although active electricalcompensation circuits, by definition, increase system power consumption,a combination of passive and active compensation (e.g., DDS resonator incombination with active compensation circuitry) may enable stabilitytargets to be achieved with less power than active compensation alone,or enable greater stability than could be achieved with either approachalone. Passive mechanical temperature compensation is possible throughmaterial selection and structure design in both homogenous and compositeresonator structures.

DDS resonators as disclosed herein may be combined with additionalelements to form a system with improved temperature stability and/orother useful functionality. In FIG. 3A, for example, a DDS resonator 301is combined with a sustaining circuit 303 to form an oscillator. Theresonant frequency of the oscillator may be modified in atemperature-dependent manner (i.e., as shown in FIG. 3A, a temperaturesignal from sensor 305 is received within frequency modifying element307 which, in turn, provides a temperature-dependent aresonant-frequency control signal to DDS resonator 301 and/or sustainingcircuit 303) and to yield a resonant frequency with improved temperaturestability compared to that achievable with the DDS resonator alone. Forexample, a temperature-dependent electrostatic field may be applieddirectly to the DDS resonator using control electrodes that areco-fabricated with the resonator. An example of this approach is shownin FIG. 3B, with electrodes formed by the DDSi layer (electrode 2) andconductive layer (electrode 1) of the three-layer resonator embodimentdescribed above (i.e., having DDSi, AN and conductive layers, thoughother resonator structures/materials may be used in alternativeembodiments), and the temperature-dependent electrostatic field formedby the time-varying and temperature-dependent difference between thepotentials at the two electrodes, Ve1-Ve2.

As another example, if one component of the resonator has a significantpiezoelectric response, the mechanical stress on the resonator can bemodified in a temperature-dependent manner to adjust the resonatorfrequency. FIG. 3C illustrates an example of such an arrangement, againin the context of the tri-layer DDSi/AlN/conductive-material resonatordiscussed above, though other resonator structures and/or materials maybe used.

In another embodiment, shown for example in FIG. 3D, a capacitiveelement may be included in a resonator sustaining circuit 321 and itscapacitance can be modified in a temperature-dependent manner, therebyeffecting a variable capacitance element 323 that may be used, forexample, to tune the frequency of the oscillator system (i.e., systemincluding DDS resonator 301 and sustaining circuit 321).

In an alternative embodiment, shown in FIG. 3E, the resonant frequencyof DDS resonator 301 is modified after being output from an oscillator(formed at least in part by the DDS resonator 301 and sustaining circuit303, as shown) by a frequency modifying element 331 within the resonatorsystem. In a more specific implementation, shown for example in FIG. 3F,a temperature-dependent signal (e.g., from a temperature sensor as shownat 305 in FIG. 3E) can be provided to a fractional-N phase locked loop(an example of a frequency modifying element) in order to generate anoutput frequency that is substantially more stable over temperature thanin the case of the resonator alone.

As shown in FIG. 3G, any of the oscillator systems shown in FIGS. 3A-3F(or other oscillator systems employing a DDS resonator) may includeprogrammable/storage circuitry 350 in which TCF coefficients, controlsettings or other information may be stored. TCF coefficients recordedor stored in such an oscillator system may be based on thecharacteristics of an individual resonator (301) or group of resonators.Additionally, the coefficients may be based on characteristics of anindividual sustaining circuit (303) or group of sustaining circuits. Forexample, the TCF behavior of oscillators or resonators may be determinedby sweeping temperature and recording frequency as shown at 355, or bymeasuring room temperature characteristics (e.g., resistivity,frequency, etc.) that are predictive of temperature-dependent behavior.

Programmable/storage circuitry in which the TCF coefficients or otherparameters indicative of temperature-dependent behavior may be storedmay include any practicable on-chip or off-chip memory elements such as,for example and without limitation, registers or other volatile ornon-volatile memory elements including, without limitationone-time-programmable (OTP) memory, electrically programmableread-only-memory (EPROM), Flash memory and so forth.

The TCF coefficients, or other parameters indicative of temperaturedependent behavior, may also be stored as a resistance value. As shownin FIG. 3H, for example, the MEMS resonator die may contain at least oneresistor that may be trimmed in resistance by a laser (or other method,such as thermal fusing) to record a value related to the behavior of thedevice over temperature. A thermistor and/or heater may also be used tostore information in this manner. Additional bondpads or electricalinterconnects are avoided by using a thermistor, or heater, for example,for this purpose.

The stored temperature behavior information may be used by an oscillatorsystem to improve its frequency stability. The system can read out thestored information, combine it with a temperature measurement, and applya correction to the resonator or oscillator system frequency. Thisoperation is shown, for example, in the embodiment of FIG. 3I, in whicha frequency-modifying element 375 receives a temperature signal fromsensor 305 and reads out temperature-based correction data (“data”) fromprogrammable storage 350. Frequency correction signals corresponding tothe correction data are then output to a resonator, sustaining circuitand/or output conditioning circuitry (e.g., PLL as discussed above inreference to FIG. 3F).

FIG. 4A illustrates an embodiment of MEMS arrangement 121 having a DDSresonator 401 integrated with one or more components of an activetemperature compensation system. The DDS resonator may be implemented inaccordance with any of the resonator embodiments described herein andincludes a layer (or other deposition or arrangement) of degeneratelydoped semiconductor material. In the embodiments shown, the temperaturecompensation circuitry includes one or more temperature sensing elements403 (e.g., thermistor or other temperature sensor), one or more heatingelements 405 and temperature control circuitry 407 to control operationof the heating elements (e.g., powering the heating elements asnecessary to reach a predetermined or programmed temperature setpoint ortemperature range). FIG. 4B illustrates an example of this temperaturecompensation operation, showing a time-varying heater output generatedin accordance with a time-varying ambient temperature to maintain aconstant or near-constant resonator temperature.

As FIG. 4A shows, integration of the DDS resonator and components of thetemperature compensation system may vary from implementation toimplementation. In one embodiment, shown for example in shaded region411, DDS resonator 401 is integrated (i.e., co-fabricated with orotherwise formed on the same substrate or die) with a temperature sensor403 (“Integrated T-Sense”), while temperature control circuitry 407 andheater 405 (i.e., one or more heating elements) are implemented off die.In another embodiment, indicated by region 415, temperature controlcircuitry 407 and heater 405 are integrated with temperature sensor 403and DDS resonator 401, thus establishing a fully integrated activetemperature compensation system within the resonator die. Though notspecifically shown, DDS resonator 401 may alternatively be integratedwith heater 405 while the temperature sensor(s) and/or temperaturecontrol circuitry remain off-die.

FIG. 5A illustrates an embodiment of a DDS resonator 500 having a numberof the features described above. More specifically, DDS resonator 500may be a single-layer structure (i.e., consisting only of a singledegenerately doped semiconductor, such as degenerately doped silicon) ora multi-layer structure having, for example, aluminum nitride (AlN) orother piezoelectric layer disposed between a degenerately doped siliconlayer and conductive layer as shown in FIG. 1I. Also, the dopant type orconcentration within the DDS layer or any other layer of resonator 500may be non-uniform (e.g., higher or lower concentration with high stressarea 501 or other areas of the resonator body), and the resonator may befabricated such that the resonator axis 503 is disposed at a nonzeroangle, ϕ (i.e., the “resonator angle”), with respect to thecrystallographic axis 504. As explained above, the resonator angle, DDSdopant concentration and type (including any non-uniformity) may each bespecifically engineered to null first-order and at least onehigher-order temperature coefficients of frequency. The mode shape,relative layer thicknesses, dopant types/concentrations of other layers(e.g., conductive layer formed from degenerately doped polysilicon) mayalso be specifically chosen, in combination with the dopantconcentration/type of the bulk DDS layer, to yield a desiredtemperature-dependent resonant behavior, such as a temperatureindicative behavior, temperature-stable (or temperature-insensitive)behavior, etc., over one or more desired operating temperature ranges.In the particular example shown, DDS resonator 500 has an ellipsoidshape with a pinched transverse dimension (i.e., orthogonal to resonatoraxis 503) between spring-bearing anchors 507 a and 507 b. Trenching 512is etched or otherwise formed around and/or beneath the resonator andanchor spring elements to release the DDS resonator and anchor springelements from (i.e., free those elements to move relative to) substratefield area 510.

FIG. 5B illustrates an exemplary cross-section of DDS resonator 500 vialine A-A in FIG. 5A, showing a degenerately-doped (DD) single-crystalsilicon layer, an aluminum nitride piezoelectric layer and adegenerately-doped polysilicon electrode layer. Spring elements(“spring”) and field area anchors are disposed on either side of theresonator body to form the respective anchors shown at 507 a and 507 bin FIG. 5A. Various other anchoring arrangements, with or without springmounts, may be employed in alternative embodiments, includingsingle-anchor arrangements or configurations of more than two anchors.

FIG. 5C illustrates an exemplary finite-element model of the DDSresonator of FIG. 5A, showing displacement and stress distributionduring resonant vibration (or oscillation). For example, vectors(arrows) projecting from the edge of the resonator body illustrate adirection of motion of the resonator during an expansion phase of anoscillation cycle. In the example shown, a high stress area occursbetween the anchors, while low stress areas occur at opposite ends ofthe resonator axis (i.e., the axis shown in FIG. 5A), and equi-stresscontours occur between the low-stress and high-stress areas. The stressin the anchors is near zero.

The following are at least some of the advantages that may be realizedby embodiments disclosed herein:

-   -   Degenerately doped silicon can replace two separate materials        used in piezoelectric resonators to date: a temperature        compensation material (e.g. SiO₂) and an electrically conductive        material (e.g. Mo). Degenerately doped silicon is capable of        temperature compensation and provides sufficiently low        electrical resistance (e.g. 1-50 ohms) to serve as an electrode        material for many applications.    -   Degenerately doped single crystal silicon facilitates improved        frequency stability over temperature by allowing cancellation of        both first- and second-order TCF. The doping and orientation of        the degenerately doped silicon layer provide at least two        degrees of freedom for cancelling at least two temperature        coefficients. Examples discussed above demonstrate that first-        and second-order TCF cancellation of a composite piezoelectric        resonator utilizing highly doped silicon is possible and is in        the accessible design space. The temperature coefficient        improvements enabled by the disclosed invention lead to the        possibility of less than +/−10 ppm of frequency variation across        the industrial temperature range from −40 to +85 C. In contrast,        micromechanical resonators with only first-order TCF        compensation typically show +/−50 to 200 ppm variation over the        industrial temperature range.    -   Elimination of the oxide and metal interfaces improves the        mechanical quality factor of the resonator through the        elimination of layers with potentially high acoustic loss and        the elimination of interfaces which each may significantly        increase the mechanical dissipation of the resonator.    -   Furthermore, replacement of the metal and oxide layers in the        structure with a semiconductor eliminates work hardening,        fatigue effects and interface effects that contribute to        frequency hysteresis over temperature and frequency aging over        time.    -   A resonator with engineered TCF characteristics may be created        out of one or more layers of material. In one layer, it is        possible to cancel the first and second order TCF coefficients        by design and doping, if the material has qualities similar to        degenerately doped silicon. There may be one or more regions in        the layer exhibiting one or more material properties. For        example, a single silicon layer may possess at least one region        containing at least one doping level and at least one dopant. A        single material layer may possess at least one region containing        at least one material type. Each of these regions may add an        additional degree of freedom in the resonator TCF behavior. By        adjusting the size and characteristics of these regions, it is        possible to affect the nominal frequency and, the first, second,        third, and higher order TCF terms.

The various MEMS systems, devices, structures, components disclosedherein, as well as related circuitry (e.g., sustaining circuitry, sensecircuitry, drive circuitry, conditioning circuitry, control circuitry,etc.) may be described using computer aided design tools and expressed(or represented), as data and/or instructions embodied in variouscomputer-readable media, in terms of their behavioral, registertransfer, logic component, transistor, layout geometries, and/or othercharacteristics. Formats of files and other objects in which suchstructure and/or circuit expressions may be implemented include, but arenot limited to, formats supporting behavioral languages such as C,Verilog, VHDL, and Matlab, formats supporting register level descriptionlanguages like RTL, and formats supporting geometry descriptionlanguages such as GDSII, GDSIII, GDSIV, CIF, MEBES and any othersuitable formats and languages. Computer-readable media in which suchformatted data and/or instructions may be embodied include, but are notlimited to, non-volatile storage media in various forms (e.g., optical,magnetic or semiconductor storage media) and carrier waves that may beused to transfer such formatted data and/or instructions throughwireless, optical, or wired signaling media or any combination thereof.Examples of transfers of such formatted data and/or instructions bycarrier waves include, but are not limited to, transfers (uploads,downloads, e-mail, etc.) over the Internet and/or other computernetworks via one or more data transfer protocols (e.g., HTTP, FTP, SMTP,etc.).

When received within a computer system via one or more computer-readablemedia, such data and/or instruction-based expressions of the abovedescribed structures, circuits and/or circuitry may be processed by aprocessing entity (e.g., one or more processors) within the computersystem in conjunction with execution of one or more other computerprograms including, without limitation, net-list generation programs,place and route programs and the like, to generate a representation orimage of a physical manifestation of such structures, circuits and/orcircuitry. Such representation or image may thereafter be used in devicefabrication, for example, by enabling generation of one or more masksthat are used to form various components of the circuits in a devicefabrication process.

Moreover, the various structures (for example, the structures of theMEMS device), circuits and/or circuitry disclosed herein may berepresented via simulations using computer aided design and/or testingtools. The simulation of the various structures and/or characteristicsor operations thereof may be implemented by a computer system whereincharacteristics and operations of such structures and/or circuitry, andtechniques implemented thereby, are imitated, replicated and/orpredicted via a computer system. The disclosed embodiments encompasssuch simulations of the exemplary structures and circuitry disclosedherein, and/or techniques implemented thereby.

In the foregoing description and in the accompanying drawings, specificterminology and drawing symbols have been set forth to provide athorough understanding of the disclosed embodiments. In some instances,the terminology and symbols may imply specific details that are notrequired to practice those embodiments. For example, any of the specificnumbers of signal path widths, signaling or operating frequencies,component circuits or devices, material types, dopant types andconcentrations and the like can be different from those described abovein alternative embodiments. Additionally, links or other interconnectionbetween integrated circuit devices or internal circuit elements orblocks may be shown as buses or as single signal lines. Each of thebuses can alternatively be a single signal line, and each of the singlesignal lines can alternatively be buses. The term “coupled” is usedherein to express a direct connection as well as a connection throughone or more intervening circuits or structures. Integrated circuitdevice “programming” can include, for example and without limitation,loading a control value into a register or other storage circuit withinthe integrated circuit device in response to a host instruction (andthus controlling an operational aspect of the device and/or establishinga device configuration) or through a one-time programming operation(e.g., blowing fuses within a configuration circuit during deviceproduction), and/or connecting one or more selected pins or othercontact structures of the device to reference voltage lines (alsoreferred to as strapping) to establish a particular device configurationor operation aspect of the device. The terms “exemplary” and“embodiment” are used to express an example, not a preference orrequirement. Also, the terms “may” and “can” are used interchangeably todenote optional (permissible) subject matter. The absence of either termshould not be construed as meaning that a given feature or technique isrequired.

The section headings in the above detailed description have beenprovided for convenience of reference only and in no way define, limit,construe or describe the scope or extent of the corresponding sectionsor any of the embodiments presented herein. Also, various modificationsand changes can be made to the embodiments presented herein withoutdeparting from the broader spirit and scope of the disclosure. Forexample, features or aspects of any of the embodiments can be applied incombination with any other of the embodiments or in place of counterpartfeatures or aspects thereof. Accordingly, the specification and drawingsare to be regarded in an illustrative rather than a restrictive sense.

What is claimed is:
 1. A microelectromechanical system (MEMS) resonatorcomprising: a degenerately-doped single-crystal silicon layer; apiezoelectric material layer disposed on the degenerately-dopedsingle-crystal silicon layer; and an electrically-conductive materiallayer (i) disposed on the piezoelectric material layer opposite thedegenerately-doped single-crystal silicon layer, and (ii) patterned toform first and second electrodes that are electrically isolated from oneanother; wherein the conductive material layer comprises heavily dopedpolysilicon.
 2. The MEMS resonator of claim/wherein a resonator axisalong which the MEMS resonator exhibits a predominant motion duringresonant oscillation is offset from a dominant crystallographic axis ofthe degenerately-doped single-crystal silicon layer by an angle thatsubstantially reduces at least one of first-order or second-ordertemperature coefficients of frequency (TCFs) of the MEMS resonatorrelative to first- and/or second-order TCFs that would result withoutangular offset between the resonator axis and the dominantcrystallographic axis.
 3. The MEMS resonator of claim/wherein thedegenerately-doped single-crystal silicon layer and the piezoelectricmaterial layer have respective thicknesses in a predetermined ratio thatattenuates at least one temperature coefficient of frequency of the MEMSresonator.
 4. The MEMS resonator of claim 1 wherein at least one offirst-order or second-order temperature coefficients of frequency (TCFs)of the degenerately-doped single-crystal silicon layer and thepiezoelectric material layer are opposite in sign over at least part ofan operating temperature range.
 5. A microelectromechanical system(MEMS) resonator comprising: a degenerately-doped single-crystal siliconlayer; a piezoelectric material layer disposed on the degenerately-dopedsingle-crystal silicon layer; and an electrically-conductive materiallayer (i) disposed on the piezoelectric material layer opposite thedegenerately-doped single-crystal silicon layer, and (ii) patterned toform first and second electrodes that are electrically isolated from oneanother; wherein at least one of first-order or second-order temperaturecoefficients of frequency (TCFs) of the degenerately-dopedsingle-crystal silicon layer and the piezoelectric material layer areopposite in sign over at least part of an operating temperature range.6. The MEMS resonator of claim 5 wherein the conductive material layercomprises metal.
 7. The MEMS resonator of claim 5 wherein thepiezoelectric material layer comprises aluminum nitride.
 8. The MEMSresonator of claim/wherein a resonator axis along which the MEMSresonator exhibits a predominant motion during resonant oscillation isoffset from a dominant crystallographic axis of the degenerately-dopedsingle-crystal silicon layer by an angle that substantially reduces atleast one of first-order or second-order temperature coefficients offrequency (TCFs) of the MEMS resonator relative to first- and/orsecond-order TCFs that would result without angular offset between theresonator axis and the dominant crystallographic axis.
 9. The MEMSresonator of claim/wherein the degenerately-doped single-crystal siliconlayer and the piezoelectric material layer have respective thicknessesin a predetermined ratio that attenuates at least one temperaturecoefficient of frequency of the MEMS resonator.
 10. Amicroelectromechanical system (MEMS) resonator comprising: adegenerately-doped single-crystal silicon layer; a piezoelectricmaterial layer disposed on the degenerately-doped single-crystal siliconlayer; and an electrically-conductive material layer (i) disposed on thepiezoelectric material layer opposite the degenerately-dopedsingle-crystal silicon layer, and (ii) patterned to form first andsecond electrodes that are electrically isolated from one another;wherein a resonator axis along which the MEMS resonator exhibits apredominant motion during resonant oscillation is offset from a dominantcrystallographic axis of the degenerately-doped single-crystal siliconlayer by an angle that substantially reduces at least one of first-orderor second-order temperature coefficients of frequency (TCFs) of the MEMSresonator relative to first- and/or second-order TCFs that would resultwithout angular offset between the resonator axis and the dominantcrystallographic axis.
 11. The MEMS resonator structure of claim 10wherein the first electrode constitutes a drive electrode for drivingthe MEMS resonator into mechanically resonant oscillation, wherein thesecond electrode comprises a sense electrode for sensing themechanically resonant oscillation of the MEMS resonator, and wherein thedegenerately-doped single crystal silicon layer constitutes a thirdelectrode.
 12. The MEMS resonator of claim 10 wherein the firstelectrode constitutes a drive electrode for driving the MEMS resonatorinto mechanically resonant oscillation, wherein the second electrodecomprises a sense electrode for sensing the mechanically resonantoscillation of the MEMS resonator, and wherein the degenerately-dopedsingle crystal silicon layer constitutes a third electrode.
 13. The MEMSresonator of claim/wherein the degenerately-doped single-crystal siliconlayer and the piezoelectric material layer have respective thicknessesin a predetermined ratio that attenuates at least one temperaturecoefficient of frequency of the MEMS resonator.
 14. Amicroelectromechanical system (MEMS) resonator comprising: adegenerately-doped single-crystal silicon layer; a piezoelectricmaterial layer disposed on the degenerately-doped single-crystal siliconlayer; and an electrically-conductive material layer (i) disposed on thepiezoelectric material layer opposite the degenerately-dopedsingle-crystal silicon layer, and (ii) patterned to form first andsecond electrodes that are electrically isolated from one another;wherein the degenerately-doped single-crystal silicon layer and thepiezoelectric material layer have respective thicknesses in apredetermined ratio that attenuates at least one temperature coefficientof frequency of the MEMS resonator.
 15. The MEMS resonator ofclaim/wherein the first electrode constitutes a drive electrode fordriving the MEMS resonator into mechanically resonant oscillation,wherein the second electrode comprises a sense electrode for sensing themechanically resonant oscillation of the MEMS resonator, and wherein thedegenerately-doped single crystal silicon layer constitutes a thirdelectrode.
 16. A microelectromechanical system (MEMS) resonatorcomprising: a degenerately-doped single-crystal silicon layer; apiezoelectric material layer disposed on the degenerately-dopedsingle-crystal silicon layer; and an electrically-conductive materiallayer (i) disposed on the piezoelectric material layer opposite thedegenerately-doped single-crystal silicon layer, and (ii patterned toform first and second electrodes that are electrically isolated from oneanother; wherein first-order and second-order temperature coefficientsof frequency (TCFs) of the MEMS resonator and at least one TCF of theMEMS resonator beyond second-order are substantially attenuated byvirtue of (i) angular offset between a dominant crystallographic axis ofthe degenerately-doped single-crystal silicon layer and a resonator axisalong which the MEMS resonator exhibits a predominant motion duringresonant oscillation, (ii) a chosen dopant concentration of thedegenerately-doped single-crystal silicon layer and (iii) chosenrelative thicknesses of the degenerately-doped single-crystal siliconlayer and piezoelectric material layer.
 17. A method of fabricatingmicroelectromechanical system (MEMS) resonator, the method comprising:disposing a piezoelectric material layer on a degenerately-dopedsingle-crystal silicon layer; disposing an electrically-conductivematerial layer on the piezoelectric material layer opposite thedegenerately-doped single-crystal silicon layer; and patterning theelectrically-conductive material layer to form first and secondelectrodes that are electrically isolated from one another; whereindegenerately doping the single-crystal silicon substrate to form thedegenerately-doped single-crystal silicon layer comprises doping thesingle-crystal silicon substrate with impurity concentration sufficientto form, as at least part of the degenerately-dopedsingle-crystal-silicon layer, a third electrode.
 18. The method of claim17 wherein patterning the electrically-conductive material layer to formthe first and second electrodes wherein the first electrode comprises:patterning a drive electrode for driving the MEMS resonator intomechanically resonant oscillation; and patterning a sense electrode forsensing the mechanically resonant oscillation of the MEMS resonator. 19.The method of claim 17 wherein disposing the electrically-conductivematerial layer on the piezoelectric material layer comprises disposing ametal layer on the piezoelectric material layer.
 20. A method offabricating microelectromechanical system (MEMS) resonator, the methodcomprising: disposing a piezoelectric material layer on adegenerately-doped single-crystal silicon layer: disposing anelectrically-conductive material layer on the piezoelectric materiallayer opposite the degenerately-doped single-crystal silicon layer; andpatterning the electrically-conductive material layer to form first andsecond electrodes that are electrically isolated from one another;wherein disposing the electrically-conductive material layer on thepiezoelectric material layer comprises disposing a heavily dopedpolysilicon layer on the piezoelectric material layer.
 21. A method offabricating microelectromechanical system (MEMS) resonator, the methodcomprising: disposing a piezoelectric material layer on adegenerately-doped single-crystal silicon layer; disposing anelectrically-conductive material layer on the piezoelectric materiallayer opposite the degenerately-doped single-crystal silicon layer; andpatterning the electrically-conductive material layer to form first andsecond electrodes that are electrically isolated from one another;wherein the method further comprises offsetting a resonator axis alongwhich the MEMS resonator exhibits a predominant motion during resonantoscillation from a dominant crystallographic axis of thedegenerately-doped single-crystal silicon layer by an angle thatsubstantially reduces at least one of first-order or second-ordertemperature coefficients of frequency (TCFs) of the MEMS resonatorrelative to first-and/or second-order TCFs that would result withoutangular offset between the resonator axis and the dominantcrystallographic axis.
 22. The method of claim 21 wherein disposing thepiezoelectric material layer on the degenerately-doped single-crystalsilicon layer comprises forming the piezoelectric material layer with athickness that yields a predetermined thickness ratio between thepiezoelectric material layer and degenerately-doped single-crystalsilicon layer that attenuates at least one temperature coefficient offrequency of the MEMS resonator.
 23. A method of fabricatingmicroelectromechanical system (MEMS) resonator, the method comprising:disposing a piezoelectric material layer on a degenerately-dopedsingle-crystal silicon layer: disposing an electrically-conductivematerial layer on the piezoelectric material layer opposite thedegenerately-doped single-crystal silicon layer; and patterning theelectrically-conductive material layer to form first and secondelectrodes that are electrically isolated from one another; whereindisposing the piezoelectric material layer on the degenerately-dopedsingle-crystal silicon layer comprises disposing an aluminum nitridelayer on the degenerately-doped single-crystal silicon layer.
 24. Themethod of claim 23 wherein degenerately doping the single-crystalsilicon substrate to form the degenerately-doped single-crystal siliconlayer comprises doping the single-crystal silicon substrate with dopantin a manner that causes at least one of first-order or second-ordertemperature coefficients of frequency (TCFs) of the degenerately-dopedsingle-crystal silicon layer and the piezoelectric material layer to beopposite in sign over at least part of an operating temperature range.25. A method of fabricating microelectromechanical system (MEMS)resonator, the method comprising: disposing a piezoelectric materiallayer on a degenerately-doped single-crystal silicon layer; disposing anelectrically-conductive material layer on the piezoelectric materiallayer opposite the degenerately-doped single-crystal silicon layer; andpatterning the electrically-conductive material layer to form first andsecond electrodes that are electrically isolated from one another;wherein disposing the piezoelectric material layer on thedegenerately-doped single-crystal silicon layer comprises forming thepiezoelectric material layer with a thickness that yields apredetermined thickness ratio between the piezoelectric material layerand degenerately-doped single-crystal silicon layer that attenuates atleast one temperature coefficient of frequency of the MEMS resonator.